Problem: Artemis seeks knowledge of the width of Orion's Belt, which is a pattern of stars in the Orion constellation. She has previously discovered the distances from her house to Alnitak $(736$ light years, $\text{l.y.})$ and to Mintaka $(915\text{ l.y.})$, which are the endpoints of Orion's Belt. She also knows the angle between these stars in the sky is $3^\circ$. What is the width of Orion's Belt? That is, what is the distance between Alnitak and Mintaka? Do not round during your calculations. Round your final answer to the nearest light year.
Explanation: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AB=d$. $\;\;3^\circ$ $d$ $915\text{ l.y.}$ $736\text{ l.y.}$ $A$ $B$ $C$ Since we are given two side lengths and the angle measure between them, we can use the law of cosines. Using the law of cosines $\begin{aligned} (AB)^2&=(AC)^2+(BC)^2-2AC\!\cdot\! BC\!\cdot\!\cos(C)\\\\ d^2&=736^2+915^2-2\cdot 736\cdot 915\cdot\cos(3^\circ) \gray{\text{Substitute}}\\\\ d&=\sqrt{736^2+915^2-2\cdot 736\cdot 915\cdot\cos(3^\circ)}\\\\ d&\approx 184 \end{aligned}$ The answer The width of Orion's belt is $184$ light years.